Method and system for utilizing space-time codes for block fading channels

ABSTRACT

A communication system for transmitting encoded signals over a communication channel is disclosed. The system includes a transmitter, which has a source that is configured to output a message signal, and an encoder that is configured to generate a code word in response to the message signal. The code word is based upon a stacking construction that is generalized for the communication channel. The communication channel is characterized as a multi-input multi-output (MIMO) block fading channel. The transmitter also includes a modulator that is configured to modulate the code word for transmission over the communication channel. Further, the transmitter includes a plurality of transmit antennas that are configured to transmit the modulated code word over the communication channel. The system also includes a receiver, which has a plurality of receive antennas. The receiver is configured to receive the transmitted code word via the plurality of receive antennas.

CROSS-REFERENCES TO RELATED APPLICATION

This application is a continuation of U.S. patent application Ser. No.10/011,831, filed Nov. 5, 2001, entitled “Method and System forUtilizing Space-Time Codes for Blocking Fading Channels”. Thisapplication is related to, and claims the benefit of the earlier filingdate of U.S. Provisional Patent Application Ser. No. 60/246,025, filedNov. 6, 2000, entitled “Method and Constructions for Space-Time Codesfor Block Fading Channels,” the entirety of which is incorporated hereinby reference.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to coding in a communication system, andis more particularly related to space-time codes having spatialdiversity and temporal diversity.

2. Discussion of the Background

Given the constant demand for higher system capacity of wirelesssystems, multiple antenna systems have emerged to increase systembandwidth vis-à-vis single antenna systems. In multiple antenna systems,data is parsed into multiple streams, which are simultaneouslytransmitted over a corresponding quantity of transmit antennas. At thereceiving end, multiple receive antennas are used to reconstruct theoriginal data stream. To combat the detrimental effects of thecommunication channel, communication engineers are tasked to developchannel codes that optimize system reliability and throughput in amultiple antenna system.

To minimize the effects of the communication channel, which typically isRayleigh, space-time codes have been garnered significant attention.Rayleigh fading channels introduce noise and attenuation to such anextent that a receiver may not reliably reproduce the transmitted signalwithout some form of diversity; diversity provides a replica of thetransmitted signal. Space-time codes are two dimensional channel codesthat exploit spatial transmit diversity, whereby the receiver canreliably detect the transmitted signal. Conventional designs ofspace-time codes have focused on maximizing spatial diversity inquasi-static fading channels and fast fading channels. However, realcommunication systems exhibit channel characteristics that are somewherebetween quasi-static and fast fading. Accordingly, such conventionalspace-time codes are not optimized.

Further, other approaches to space-time code design assume that channelstate information (CSI) are available at both the transmitter andreceiver. Thus, a drawback of such approaches is that the designrequires the transmitter and receiver to have knowledge of the CSI,which increases implementation costs because of the need for additionalhardware. Moreover, these approaches view the transmit diversityattending the use of space-time codes as a substitute for timediversity; consequently, such space-time codes are not designed to takeadvantage of other forms of diversity.

Based on the foregoing, there is a clear need for improved approachesfor providing space-time codes that can be utilized in a multi-inputmulti-output (MIMO) block fading channel. There is also a need to designspace-time codes that can exploit spatial diversity as well as timediversity. There is also a need to improve system reliability withoutreducing transmission rate. There is a further need to simplify thereceiver design. Therefore, an approach for constructing space-timecodes that can enhance system reliability and throughput in a multipleantenna system is highly desirable.

SUMMARY OF THE INVENTION

The present invention addresses the above stated needs by providingspace-time codes to optimally exploit the spatial and temporal diversityavailable in the multi-input multi-output (MIMO) block fading channels.

According to one aspect of the invention, a method for transmittingencoded signals over a communication channel of a communication systemis provided. The method includes receiving a message signal, andgenerating a code word in response to the message signal. The code wordis based upon a stacking construction that is generalized for thecommunication channel. The communication channel is characterized as amulti-input multi-output block fading channel. Under this approach,spatial diversity and temporal diversity are enhanced, withoutsacrificing transmission rate.

According to another aspect of the invention, an apparatus for encodingsignals for transmission over a communication channel of a communicationsystem is provided. The apparatus comprises a source that is configuredto output a message signal, and an encoder that is configured togenerate a code word in response to the message signal. The code word isbased upon a stacking construction that is generalized for thecommunication channel. The communication channel is characterized as amulti-input multi-output block fading channel. The above arrangementadvantageously improves system throughput and system reliability of acommunication system.

According to one aspect of the invention, an apparatus for encodingsignals for transmission over a communication channel of a communicationsystem is provided. The apparatus includes means for receiving a messagesignal, and means for generating a code word in response to the messagesignal. The code word is based upon a stacking construction that isgeneralized for the communication channel. The communication channel ischaracterized as a multi-input multi-output block fading channel. Theabove arrangement advantageously provides increased system capacity.

According to another aspect of the invention, a communication system fortransmitting encoded signals over a communication channel is disclosed.The system includes a transmitter, which has a source that is configuredto output a message signal, and an encoder that is configured togenerate a code word in response to the message signal. The code word isbased upon a stacking construction that is generalized for thecommunication channel. The communication channel is characterized as amulti-input multi-output block fading channel. The transmitter alsoincludes a modulator that is configured to modulate the code word fortransmission over the communication channel. Further, the transmitterincludes a plurality of transmit antennas that are configured totransmit the modulated code word over the communication channel. Thesystem also includes a receiver, which has a plurality of receiveantennas; the receiver is configured to receive the transmitted codeword via the plurality of receive antennas. The above arrangementadvantageously maximizes spatial and temporal diversity.

According to another aspect of the invention, a waveform signal fortransmission over a communication channel of a communication system isdisclosed. The waveform signal includes a code word that is based upon astacking construction, which is generalized for the communicationchannel. The communication channel is characterized as a multi-inputmulti-output block fading channel. The above approach minimizes datatransmission errors.

In yet another aspect of the invention, a computer-readable mediumcarrying one or more sequences of one or more instructions fortransmitting encoded signals over a communication channel of acommunication system is disclosed. The one or more sequences of one ormore instructions include instructions which, when executed by one ormore processors, cause the one or more processors to perform the step ofreceiving a message signal. Another step includes generating a code wordin response to the message signal. The code word is based upon astacking construction that is generalized for the communication channel.The communication channel is characterized as a multi-input multi-outputblock fading channel. This approach advantageously provides simplifiedreceiver design.

In yet another aspect of the present invention, an apparatus forreceiving signals over a communication channel of a communication systemis provided. The apparatus includes a demodulator that is configured todemodulate a signal containing a code word, wherein the code word beingbased upon a stacking construction that is generalized for thecommunication channel. The communication channel is characterized as amulti-input multi-output block fading channel. The apparatus alsoincludes a decoder that is configured to decode the code word and tooutput a message signal. Under this approach, the effective bandwidth ofthe communication system is increased.

BRIEF DESCRIPTION OF THE DRAWINGS

A more complete appreciation of the invention and many of the attendantadvantages thereof will be readily obtained as the same becomes betterunderstood by reference to the following detailed description whenconsidered in connection with the accompanying drawings, wherein:

FIG. 1 is a diagram of a communication system configured to utilizespace-time codes, according to an embodiment of the present invention;

FIG. 2 is a diagram of an encoder that generates space-time codes, inaccordance with an embodiment of the present invention;

FIG. 3 is a diagram of a decoder that decodes space-time codes,according to an embodiment of the present invention;

FIG. 4 is a flow chart of the process of constructing space-time codesused in the system of FIG. 1;

FIG. 5 is a diagram of a wireless communication system employing thespace-time codes, according to an embodiment of the present invention;and

FIG. 6 is a diagram of a computer system that can perform the processesof encoding and decoding of space-time codes, in accordance with anembodiment of the present invention.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

In the following description, for the purpose of explanation, specificdetails are set forth in order to provide a thorough understanding ofthe invention. However, it will be apparent that the invention may bepracticed without these specific details. In some instances, well-knownstructures and devices are depicted in block diagram form in order toavoid unnecessarily obscuring the invention.

Although the present invention is discussed with respect to MIMO blockfading channels, the present invention has applicability tocommunication channels with other channel characteristics.

FIG. 1 shows a diagram of a communication system configured to utilizespace-time codes, according to an embodiment of the present invention. Adigital communication system 100 includes a transmitter 101 thatgenerates signal waveforms across a communication channel 103 to areceiver 105. In the discrete communication system 100, transmitter 101has a message source that produces a discrete set of possible messages;each of the possible messages have a corresponding signal waveform.These signal waveforms are attenuated, or otherwise altered, bycommunications channel 103. As a result, receiver 105 must be able tocompensate for the attenuation that is introduced by channel 103. Toassist with this task, transmitter 101 employs coding to introduceredundancies that safeguard against incorrect detection of the receivedsignal waveforms by the receiver 105.

FIG. 2 shows a diagram of an encoder that generates space-time codes, inaccordance with an embodiment of the present invention. A transmitter200, as mentioned above, possesses a message source 201 that generates ksignals from a discrete alphabet, X. Encoder 203 then generates signalsfrom alphabet Y to a modulator 205. Modulator 205 maps the encodedmessages from encoder 203 to signal waveforms that are transmitted toL_(t) number of antennas 207, which emit these waveforms over thecommunication channel 103 (as shown in FIG. 1). Accordingly, the encodedmessages are segmented into L_(t) data streams and then simultaneouslytransmitted over the antennas 207.

FIG. 3 shows a diagram of a decoder that decodes space-time codes,according to an embodiment of the present invention. At the receivingside, a receiver 300 includes a demodulator 301 that performsdemodulation of received signals from transmitter 200. These signals arereceived at multiple antennas 303. After demodulation, the receivedsignals are forwarded to a decoder 305, which attempts to reconstructthe original source messages by generating messages, X′. Receiver 300,according to one embodiment of the present invention, has a memory 307that stores channel state information (CSI) associated with thecommunication channel 103. Conventional communication systems typicallyrequire that CSI be available at both the transmitter and the receiver.By contrast, the present invention, according to one embodiment, doesnot require CSI at the transmitter 200, thus, providing a more robustdesign.

FIG. 4 shows a flow chart of the process of constructing space-timecodes used in the system of FIG. 1. As stated previously, some researchhave explored the availability of multiple transmit antennas indesigning channel codes that exploit spatial transmit diversity. Thepresent invention focuses on a quasi-static fading model of thecommunication channel 103 in which the path gains remain fixedthroughout the code word. Specifically, in the present invention, themore general block fading model is examined, based upon the quasi-staticfading channel. In the present model, the code word is composed ofmultiple blocks. The fading coefficients are constant over one fadingblock but are independent from block to block. The number of fadingblocks per code word can be regarded as a measure of the interleavingdelay allowed in the system 100, so that systems subject to a strictdelay constraint are usually characterized by a small number ofindependent blocks. The information theoretic capacity of multipleantenna systems 100 in block fading channels has been studied; assumingthe availability of channel state information (CSI) at both thetransmitter 101 and receiver 105, it is noted that antenna diversity canbe a substitute for time diversity. However, this conclusion does nothold when CSI is only available at the receiver 105. The code design ofthe present invention exploits both spatial and temporal diversity.

Spatial and temporal diversity can be achieved through the constructionof space-time codes, as described in FIG. 4. In step 401, a basebanddesign criteria that determine the diversity and coding advantageachieved over block fading channels are developed. As will be more fullydescribed, this design criteria, as in [1] [2], for quasi-static andfast fading channels are special cases of the new criteria. Then,according to step 403, the diversity advantage baseband design criterionis translated into binary rank criteria for PSK modulated codes; thisapproach is more fully detailed in a paper by A. R. Hammons Jr. and H.El Gamal, entitled “On the theory of space-time codes for PSKmodulation” IEEE Trans. Info. Theory, March 2000, which is incorporatedherein by reference in its entirety. Next, the binary rank criteria arethen used to develop an algebraic framework for space-time code designin block fading design, per step 405. Such an algebraic frameworkpermits construction of space-time codes that realize the optimumtrade-off between diversity advantage and transmission rate forarbitrary number of transmit antennas and fading blocks (step 407).According to the present invention, the notion of universal space-timecoding, which aims at constructing codes that exploit the spatial andtemporal diversity when available, is described. This notion is ageneralization of the smart greedy design principle proposed in [1].

In addition to its importance in the study of code design in timevarying fading channels, the MIMO block fading model of communicationschannel 103 is beneficial in the frequency selective scenario. Thedeveloped algebraic framework is employed to construct space-frequencycodes that exploit the frequency diversity available in the MIMOfrequency selective fading channels.

In system 100, maximum likelihood (ML) decoding at receiver 105 isassumed. This assumption represents the case in which only a smallnumber of receive antennas 303 (FIG. 3) are available so that thereceiver signal processing capabilities are limited. In the case inwhich a relatively large number of receive antennas are available, moreefficient signal processing techniques can be employed to separate thedifferent transmit antenna signals. The code design criterion isdifferent in both scenarios. The present invention develops a codingapproach for systems 100 with small number of receive antennas 303.

Before discussing the construction of space-time codes of the presentinvention, it is instructive to discuss the concepts of space-time codedesign in general. First, a system model is discussed, and then thedesign criteria for space-time code design in quasi-static and fastfading channels, respectively, are described.

The system model, according to an embodiment of the present invention,the source 201 of transmitter 200 generates k information symbols fromthe discrete alphabet X, which are encoded by the error control code Cvia encoder 203 to produce code words of length N=nL_(t) over the symbolalphabet Y. The encoded symbols are parsed among L_(t) transmit antennas207 and then mapped by the modulator into constellation points from thediscrete complex-valued signaling constellation Ω for transmissionacross the channel 103. The modulated streams for all antennas 207 aretransmitted simultaneously.

At the receiver 300, there are L_(r) receive antennas 303 to collect theincoming transmissions. The received baseband signals are subsequentlydecoded by the space-time decoder 305. Each spatial channel (the linkbetween one transmit antenna (e.g., 207) and one receive antenna (e.g.,303)) is assumed to experience statistically independent flat Rayleighfading. Receiver noise is assumed to be AWGN (Additive White GaussianNoise).

A space-time code is defined to include an underlying error control codetogether with a spatial parser. An L_(t)×n space-time code C of size Sconsists of an (L_(t)n, S) error control code C and a spatial parser σthat maps each code word vector c∈C to an L_(t)×n matrix c, whoseentries are a rearrangement of those of c. Except as noted to thecontrary, it is assumed that the standard parser maps the followingc=(c₁ ¹, c₁ ², . . . , c₁ ^(L) ^(t) , c₂ ¹, c₂ ², . . . , c₂ ^(L) ^(t) ,c_(n) ¹, c_(n) ², . . . , c_(n) ^(L) ^(t) )∈Cto the matrix

$c = \begin{bmatrix}c_{1}^{1} & c_{2}^{1} & \cdots & c_{n}^{1} \\c_{1}^{2} & c_{2}^{2} & \cdots & c_{n}^{2} \\\vdots & \vdots & ⋰ & \vdots \\c_{1}^{L_{t}} & c_{2}^{L_{t}} & \cdots & c_{n}^{L_{t}}\end{bmatrix}$In this notation, it is understood that c_(t) ^(i), is the code symbolassigned to transmit antenna i (e.g., 207) at time t.

Assuming that f:Y→Ω is the modulator mapping function, then s=f(c) isthe baseband version of the code word as transmitted across the channel103. For this system, we have the following baseband model of thereceived signal:

$y_{t}^{j} = {{\sqrt{E_{s}}{\sum\limits_{i = 1}^{L}{\alpha_{i}^{ij}s_{t}^{i}}}} + n_{t}^{j}}$where √{square root over (E_(s))} is the energy per transmitted symbol;α_(t) ^(ij) is the complex path gain from transmit antenna i (e.g., 207)to receive antenna j (e.g., 303) at time t, s_(t) ^(i)=f(c_(t) ^(i)) isthe transmitted constellation point from antenna i at time t, n_(t) ^(j)is the additive white Gaussian noise sample for receive antenna j attime t. The noise samples are independent samples of zero-mean complexGaussian random variable with variance N₀/2 per dimension. The differentpath gains α_(t) ^(ij) are assumed to be statistically independent.

The fading model of primary interest is that of a block flat Rayleighfading process in which the code word encompasses M fading blocks. Thecomplex fading gains are constant over one fading block but areindependent from block to block. The quasi-static and fast fading modelsare special cases of the block fading model in which M=1, and M=n,respectively. For simplicity, it is assumed that M divides n.

For the quasi-static flat Rayleigh fading channel, the pairwise errorprobability is given by [1] [2]:

${P( carrow e )} \leq ( \frac{1}{\prod\limits_{i = 1}^{d}( {1 + {\lambda_{i}{E_{S}/4}N_{0}}} )} )^{L_{r}}$${P( carrow e )} \leq ( \frac{\mu\; E_{s}}{4N_{0}} )^{- {dL}_{r}}$where d=rank(f(c)−f(e)) and μ=(λ₁λ₂ . . . λ_(d))^(1/d) is the geometricmean of the nonzero eigen-values of A=(f(c)−f(e))(f(c)−f(e))^(H). Theexponent dL_(t) is, also referred to as the “diversity advantage,” whilethe multiplicative factor μ is known as the “coding advantage.” Theparameter d is the diversity provided by the multiple transmit antennas207.

Accordingly, the rank and equivalent product distance criteria forspace-time codes in quasi-static channels [1] results. Specifically, therank criterion specifies that the transmit diversity advantaged=rank(f(c)−f(e)) be maximized over all pairs of distinct code words c,e∈C. Further, the product distance criterion requires the maximizationof the coding advantage μ=(λ₁λ₂ . . . λ_(d))^(1/d) over all pairs ofdistinct code words c, e∈C. The rank criterion is the more important ofthe two criteria, as it determines the asymptotic slope of theperformance curve as a function of E_(s)/N₀.

The binary rank criterion for BPSK (Binary Phase-ShiftKeying)-modulated, binary space-time codes are now developed. It isassumed that C is a linear L_(t)×n space-time code with underlyingbinary code C of length N=nL_(t), wherein n>L_(t). It is also assumedthat every non-zero code word c is a matrix of full rank over the binaryfield F. Thus, for BPSK transmission over the quasi-static fadingchannel, the space-time code C achieves full spatial diversityL_(t)L_(r).

Using the above binary rank criterion, the following generalconstruction for space-time codes, which is referred to as the “stackingconstruction” is developed. It is assumed that M₁, M₂, . . . , M_(L)_(t) are binary matrices of dimension k×n,n≧k, and C is a L_(t)×nspace-time code of dimension k consisting of the code word matrices

$c = \begin{bmatrix}{\underset{\_}{x}M_{1}} \\{\underset{\_}{x}M_{2}} \\\vdots \\{\underset{\_}{x}M_{L_{t}}}\end{bmatrix}$where x denotes an arbitrary k-tuple of information bits and L_(t)<n.Consequently, C satisfies the binary rank criterion, and thus, for BPSKtransmission over the quasi-static fading channel, achieves full spatialdiversity L_(t)L_(r), if and only if M₁, M₂, . . . , M_(L) _(t) have theproperty that∀a₁, a₂, . . . , a_(L) _(t) ∈F:M=a₁M₁⊕a₂M₂⊕ . . . ⊕a_(L) _(t) M_(L) _(t)is of full rank k unless a₁=a₂= . . . =a_(L) _(t) =0.

The previously described binary rank criteria and stacking constructionmay lift to the Z₄ domain, where they describe full diversity QPSK(Quadrature Phase-Shift Keying)-modulated space-time codes. The stackingconstruction encompasses as special cases transmit delay diversity,Tarokh's hand-crafted trellis codes [1], and certain block andconcatenated coding schemes. The stacking construction is general in thenumber of transmit antennas 207 and applies to tail terminated trelliscodes as well as block codes. The important class of rate 1/L_(t) binaryconvolutional codes with optimal d_(free) can usually be spatiallyformatted to yield full diversity space-time codes.

The present invention generalizes the stacking construction in twodifferent ways. First, a generalization is described that allows forincreasing the transmission rate at the expense of minimal reduction inthe diversity advantage. This construction can then be extended to blockfading channels in which an objective is to optimally exploit thespatial and temporal diversity available in the channel withoutcompromising the transmission rate.

With respect to a fast fading channel, the pairwise error probabilitythat the decoder 305 prefers an alternate code word e to c can be upperbounded by [1]

${P( carrow e )} \leq ( \frac{1}{\prod\limits_{i = 1}^{d}( {1 + {{{{f( {\overset{\_}{c}}_{t} )} - {f( {\overset{\_}{e}}_{t} )}}}^{2}{E_{S}/4}N_{0}}} )} )^{L_{r}}$${P( carrow e )} \leq ( \frac{\mu\; E_{s}}{4N_{0}} )^{- {dL}_{r}}$where c_(t), is the i^(th) column of c, e_(t) is the t^(th) column of e,d is the number of columns c_(t) that are different from e_(t), and

$\mu = ( {\prod\limits_{c_{t} \neq e_{t}}{{{f( {\overset{\_}{c}}_{t} )} - {f( {\overset{\_}{e}}_{t} )}}}^{2}} )^{1/d}$

The diversity advantage is dL_(r); and the coding advantage is μ. Thus,the fundamental design criteria [1] for space-time codes over fastfading channels are as follows. For the distance criterion, the numberof column differences d=|{t:{overscore (c)}_(t)≠ē_(t)}| over all pairsof distinct code words c, e∈C are maximized. In addition, for productcriterion, the following coding advantage is maximized over all pairs ofdistinct code words c, e∈C:

$\mu = ( {\prod\limits_{c_{t} \neq e_{t}}{{{f( {\overset{\_}{c}}_{t} )} - {f( {\overset{\_}{e}}_{t} )}}}^{2}} )^{1/d}$

Because real fading channels are neither quasi-static nor fast fading,but something in between, Tarokh et al. [1] suggested the ad-hocstrategy of designing space-time codes based on a combination of thequasi-static and fast fading design criteria. They refer to space-timecodes designed according to the hybrid criteria as “smart greedy codes,”meaning that the codes seek to exploit both spatial and temporaldiversity whenever available. The present invention introduces thenotion of universal space-time coding that attempts to achieve the samegoal, in a systematic fashion, under the MIMO block fading model.

The present invention considers the block fading model and develops asystematic approach for designing space-time codes that achieve themaximum diversity advantage for any coding rate, number of transmitantennas 207, and temporal interleaving depth. First, the designcriteria that governs the code performance in such channels aredeveloped. The baseband design criterion is described. Under the blockfading assumption, the path gains are constant over n/M consecutivesymbol durations. The notation (.)[m] denotes the single or twodimensional vector of values of parameter (.) for the m^(th) fadingblock. Accordingly, the following parameters are defined:α^(ij) [m]=α _((m−1n/M+1)) ^(ij)= . . . =α_(mn/M) ^(ij),Y[m]=[y _(((m−1)n/M)+1) ¹ , . . . , y _(mn/M) ¹ , . . . , y _(mn/M) ^(L)^(r) ],N[m]=[n _(((m−1)n/M)+1) ¹ , . . . , n _(mn/M) ¹ , . . . , n _(mn/M) ^(L)^(r) ],A[m]=[α ¹¹ [m], . . . , α ^(L) ^(t) ¹ [m], . . . , α ^(L) ^(t) ^(L) ^(r)[m]],c[m]=[{overscore (c)} _(((m−1)n/M)+1) , . . . , {overscore (c)}_(mn/M)],For 1≦m≦M, the following expression exists:Y [m]=√{square root over (E _(s))} A [m]D _(c)[m]+ N [m]If code word c is transmitted, then the conditional pairwise errorprobability that the decoder 305 will prefer the alternate code word eto c is given by the following:P(c→e|{α ^(ij)})=P(V<0|α^(ij)),where

$V = {\sum\limits_{m = 1}^{M}\lbrack {{{{{\underset{\_}{A}\lbrack m\rbrack}( {{D_{c}\lbrack m\rbrack} - {D_{e}\lbrack m\rbrack}} )} + {\underset{\_}{N}\lbrack m\rbrack}}}^{2} - {{\underset{\_}{N}\lbrack m\rbrack}}^{2}} \rbrack}$is a Gaussian random variable with mean

${E\{ V \}} = {\sum\limits_{m = 1}^{M}{{{{\underset{\_}{A}\lbrack m\rbrack}( {{D_{c}\lbrack m\rbrack} - {D_{e}\lbrack m\rbrack}} )} + {\underset{\_}{N}\lbrack m\rbrack}}}^{2}}$and variance

${{{Var}\{ V \}} = {2N_{0}E{\{ V \}.{Thus}}}},{{{P( {V < 0} \middle| \{ \alpha^{ij} \} )}Q} = {( \frac{\sum\limits_{n = 1}^{M}{{{\underset{\_}{A}\lbrack m\rbrack}( {{D_{c}\lbrack m\rbrack} - {D_{e}\lbrack m\rbrack}} )}}}{\sqrt{2\; N_{0}}} ) \leq {\frac{1}{2}\exp\{ {{- \frac{1}{4N_{0}}}{\sum\limits_{m = 1}^{M}{{{\underset{\_}{A}\lbrack m\rbrack}( {{D_{c}\lbrack m\rbrack} - {D_{e}\lbrack m\rbrack}} )}}^{2}}} \}}}}$Accordingly, the pairwise probability of error can be manipulated toyield the fundamental bound [1] [2]:

${P( carrow e )} \leq ( \frac{\mu\; E_{s}}{4N_{0}} )^{- {dL}_{r}}$where

${d = {\sum\limits_{m = 1}^{M}d_{m}}},{d_{m} = {{rank}( {{f( {c\lbrack m\rbrack} )} - {f( {e\lbrack m\rbrack} )}} )}},{\mu = ( {\prod\limits_{m = 1}^{M}{{\lambda_{1}\lbrack m\rbrack}{\lambda_{2\;}\lbrack m\rbrack}\mspace{14mu}\cdots\mspace{14mu}{\lambda_{d_{m}}\lbrack m\rbrack}}} )^{1/d}},$λ₁[m], λ₂[m], . . . , λ_(d) _(m) [m] are the nonzero eigenvalues of·A[m]=(f(c[m])−f(e[m]))(f(c[m])−f(e[m]))^(H). Hence, the generalizeddiversity and product distance criteria for space-time codes over blockfading channels are as follows: for the block fading baseband rankcriterion, the transmit diversity advantage

$d_{m} = {\sum\limits_{m = 1}^{M}{{rank}( {{f( {c\lbrack m\rbrack} )} - {f( {e\lbrack m\rbrack} )}} )}}$over all pairs of distinct code words c, e∈C is maximized, and the blockfading product distance criterion entails maximizing the codingadvantage

$\mu = ( {\prod\limits_{m = 1}^{M}{{\lambda_{1}\lbrack m\rbrack}{\lambda_{2}\lbrack m\rbrack}\mspace{14mu}\cdots\mspace{14mu}{\lambda_{d_{m}}\lbrack m\rbrack}}} )^{1/d}$over all pairs of distinct code words c, e∈C.

It is straightforward to observe that the design criteria forquasi-static and fast fading channels can be obtained from the blockfading generalized criteria by simply letting M=1, M=n, respectively. Asin the case of quasi-static fading channels, the fact that the basebandrank criterion applies to the complex-valued differences betweenbaseband code words represents a major obstacle to the systematic designof space-time codes that achieve the maximum possible diversityadvantage. It is possible to translate the baseband criterion into anequivalent binary rank criteria for BPSK and QPSK modulated space-timecodes. These general binary criteria are sufficient to ensure that aspace-time code achieves a certain level of diversity over block fadingchannels. To this end, a certain equivalence relation is imposed uponpotential baseband difference matrices. Each equivalence class containsa special representative that is seen to be a binary projection of acode word whose rank over the binary field is a lower bound on the rankof any of the complex matrices in the equivalence class.

Turning now to the discussion of the BPSK binary rank criterion, forBPSK modulation, the natural discrete alphabet is the field F={0,1} ofintegers modulo 2. Modulation is performed by mapping the symbol x∈F tothe constellation point s=f(x)∈{−1,1} according to the rule s=(−1)^(x).It is noted that it is possible for the modulation format to include anarbitrary phase offset e^(iφ), since a uniform rotation of the BPSKconstellation will not affect the rank of the matricesf(c[m])−f(e[m])nor the eigenvalues of the matricesA[m]=(f(c[m])−f(e[m]))(f(c[m])−f(e[m])^(H). Notationally, the circledoperator ⊕ will be used to distinguish modulo 2 addition from real- orcomplex-valued (+, −) operations.

Based on the above discussion, the following result for BPSK modulatedcodes over block fading (BF) channels is achieved. It is assumed that Cis a linear L_(t)×n space-time code with n≧L_(t) used in communicationsystem 100 with L_(t) transmit antennas 207, L_(r) receive antennas 303,and operating over a block fading channel 103 with M blocks per codeword. Additionally, it is supposed that every non-zero binary code wordc∈C has the property that d is the largest integer such that

${{\sum\limits_{m = 1}^{M}{{rank}( {c\lbrack m\rbrack} )}} \geq d},$where the rank is over the binary field F. Then, for BPSK transmission,the space-time code C achieves a diversity level at least as large asdL_(r). This result stems from the fact that for any two code words c,e∈C, the following expression holds true:rank(f(c[m])−f(e[m]))≧rank(f(c[m])⊕f(e[m])),where the rank in the left hand side is over the complex field and inthe right hand side is over the binary field. Hence,

${\sum\limits_{m = 1}^{M}{{rank}( {{f( {c\lbrack m\rbrack} )} - {f( {e\lbrack m\rbrack} )}} )}} \geq {\sum\limits_{m = 1}^{M}{{rank}( {{f( {c\lbrack m\rbrack} )} \oplus {f( {e\lbrack m\rbrack} )}} )}}$Since C is linear, then c⊕e∈C and

${\min\limits_{c,{e \in C}}{\sum\limits_{m = 1}^{M}{{rank}( {{f( {c\lbrack m\rbrack} )} - {f( {e\lbrack m\rbrack} )}} )}}} \geq {\min\limits_{c \in C}{\sum\limits_{m = 1}^{M}{{{rank}( {c\lbrack m\rbrack} )}.}}}$

As regards the QPSK binary rank criterion, in QPSK modulation, thenatural discrete alphabet is the ring Z₄={0,±1,2} of integers modulo 4.Modulation is performed by mapping the symbol x∈Z₄ to the constellationpoint s∈{±1,±i} according to the rule s=i^(x),

where i=√{square root over (−1)}. Again, the absolute phase reference ofthe QPSK constellation could have been chosen arbitrarily withoutaffecting the diversity advantage or coding advantage of a Z₄-valuedspace-time code.

For the Z₄-valued matrix c[m], the binary-valued indicant projectionsare defined: Ξ(c[m]) and Ψ(c[m]). These indicant projections as definedbelow serve to indicate certain aspects of the binary structure of theZ₄ matrix in which multiples of two are ignored.

A Z₄-valued matrix c[m]of dimension L_(t)×n/M is said to be of type 1^(l) 2 ^(L) ^(t) ^(−l) if it consists of exactly l rows that are notmultiples of two.

It is assumed that c[m] is a Z₄-valued matrix of type 1 ^(l) 2 ^(L) ^(t)^(−l). After suitable row permutations if necessary, it has thefollowing row structure:

${c\lbrack m\rbrack} = \begin{bmatrix}{{\underset{\_}{c}}_{1}\lbrack m\rbrack} \\\vdots \\{{\underset{\_}{c}}_{1}\lbrack m\rbrack} \\{2{{\underset{\_}{c}}_{l + 1}^{\prime}\lbrack m\rbrack}} \\\vdots \\{2{{\underset{\_}{c}}_{L_{t}}^{\prime}\lbrack m\rbrack}}\end{bmatrix}$The row-based indicant projection (5-projection) is then defined as thefollowing:

${\Xi( {c\lbrack m\rbrack} )} = \begin{bmatrix}{\beta( {{\underset{\_}{c}}_{1}\lbrack m\rbrack} )} \\\vdots \\{\beta( {{\underset{\_}{c}}_{1}\lbrack m\rbrack} )} \\{\beta( {{\underset{\_}{c}}_{l + 1}^{\prime}\lbrack m\rbrack} )} \\\vdots \\{\beta( {{\underset{\_}{c}}_{L_{t}}^{\prime}\lbrack m\rbrack} )}\end{bmatrix}$Similarly, the column-based indicant projection (Ψ-projection) isdefined as[Ψ(c[m])]^(T)=Ξ(c[m])^(T))

Using the binary indicants, we have the following result that translatesthe baseband rank criterion into binary rank criterion for QPSKmodulated codes in a block fading environment.

With respect to the QPSK-BF binary rank criterion, it is assumed that Cis a linear L_(t)×n space-time code over Z₄, with n≧L_(t) incommunication system 100 with L_(t) transmit antennas 207, L_(r) receiveantennas 303, and operating over a block fading channel 103 with Mblocks per code word. It is further assumed that, every non-zero codeword c∈C has the property that d is the largest integer such that

${\sum\limits_{m = 1}^{M}{{rank}\{ {\Xi( {c\lbrack m\rbrack} )} \}}} \geq {d\mspace{14mu}{or}\mspace{14mu}{\sum\limits_{m = 1}^{M}{{rank}\{ {\Psi( {c\lbrack m\rbrack} )} \}}}} \geq d$where the rank is over the binary field F. Then, for QPSK transmission,the space-time code C achieves a diversity level at least as large asdL_(r).

The block fading channel binary rank criteria open the door fordeveloping an algebraic framework for constructing space-time codes thatrealize the optimum trade-off between transmission rate and diversityadvantage. First, it is shown that the design of spec-time codes thatrealize full diversity is relatively easy using the multi-stackingconstruction in [2]. However, attaining full diversity entails asignificant loss in the transmission rate compared to the quasi-staticscenario, irrespective of the coding scheme used. The more challengingtask, as recognized by the present invention, is to construct space-timecodes that exploit the temporal diversity in a multi-input multi-output(MIMO) block fading channel without compromising the transmission rate.This coding paradigm can achieve significant increase in the diversityadvantage with a relatively small number of fading blocks per code word(i.e., a small penalty in terms of interleaving delay).

The algebraic framework for BPSK space-time codes is now described. In aMIMO block fading channel 103 with L_(t) transmit antennas 207 and Mfading blocks per codeword the maximum transmit diversity is L_(t)M.Under this model, the space-time code C is defined to include thefollowing code words

$c = \begin{bmatrix}{\underset{\_}{x}\; M_{11}} & {\underset{\_}{x}\; M_{12}} & \cdots & {\underset{\_}{x}\; M_{1M}} \\{\underset{\_}{x}\; M_{21}} & {\underset{\_}{x}\; M_{22}} & \cdots & {\underset{\_}{x}\; M_{2M}} \\\vdots & \vdots & ⋰ & \vdots \\{\underset{\_}{x}\; M_{L_{t}1}} & {\underset{\_}{x}\; M_{L_{t}2}} & \cdots & {\underset{\_}{x}\; M_{L_{t}M}}\end{bmatrix}$where x∈F^(k),M_(ij)∈F^(k×n/M). Full diversity is realized in thischannel 103 by the following multi-stacking code construction. Thespace-time code C achieves full transmit diversity L_(t)M if for everyi, 1≦i≦M, the set of matrices {M_(1i), M_(2i), . . . , M_(L) _(t) _(i)}satisfies the stacking construction as discussed previously.

The main disadvantage of the above multi-stacking construction is thatthe transmission rate is reduced to 1/M bits/sec/Hz; it is noted that inquasi-static fading channels, the stacking construction supports 1bits/sec/Hz. This reduction in rate is not a special characteristic ofthe multi-stacking construction and any space-time code that achievesfull diversity in the MIMO block fading channel suffers from thisadvantage. This fact is formalized in the following result that servesto establish the fundamental limit on the transmission rate forspace-time codes with a certain level of diversity advantage in MIMOblock fading channels.

The maximum transmission rate for BPSK modulation in a communicationsystem 100 with L_(t) transmit antennas 207, operating over a blockfading channel 103 with M blocks and using a space-time code thatachieves d levels of transmit diversity is

$\frac{{ML}_{t} - d + 1}{M}\mspace{14mu}{bits}\text{/}\sec\text{/}{{Hz}.}$It is observed that the scenario at hand is strictly more restrictivethan the case of single transmit antenna with ML_(t) fading block—i.e.,a code that achieves d levels of diversity in the MIMO block fadingscenario achieves at least the same diversity order in the laterscenario.

At this point, the general case of constructing codes that realize theoptimum trade between the transmission rate and diversity advantage forany arbitrary transmission rate is described. The main result forspace-time code design in MIMO block fading channels is obtained in twosteps. First, a generalization of the stacking construction inquasi-static fading channels that allows for increasing the transmissionrate at the expense of minimal reduction the diversity advantage isachieved. This important result introduces the technical machinerynecessary for the second step, in which space-time codes are constructedto optimally exploit the diversity available in MIMO block fadingchannels.

In the first step, the previously discussed stacking construction isgeneralized to handle codes that achieve d<L_(t) levels of spatialtransmit diversity. These codes are capable of supporting highertransmission rates than full diversity codes. For example, with BPSKmodulation and L_(t) transmit antennas 207, a d-diversity code cansupport η≧1 bits/sec/Hz, where L_(t)−d+1≧η≧L_(t)−d, over thequasi-static fading channel. Before proceeding further, certaindefinitions and terminologies are presented to facilitate the discussionof the generalized stacking construction.

The BPSK space-time code under consideration C is defined as in thepreviously described stacking construction with the modification that ncan be smaller than k to allow for higher transmission rates than 1bits/sec/Hz. G is defined as the set of binary full rank matrices {G:G=└g_(i,j)┘_(L) _(t) _(xL) _(t) } resulting from applying any number ofsimple row operations to the identity matrix, I_(L) _(t) . Also,∀G∈G,x∈F^(K), let

${Q( {\underset{\_}{x},G} )} = {{G\begin{bmatrix}{\underset{\_}{x}\; M_{1}} \\{\underset{\_}{x}\; M_{2}} \\\vdots \\{\underset{\_}{x}\; M_{L_{t}}}\end{bmatrix}} = \lbrack {{q_{1}^{T}( {\underset{\_}{x},G} )},{q_{2}^{T}( {\underset{\_}{x},G} )},\ldots\mspace{14mu},{q_{L_{t}}^{T}( {\underset{\_}{x},G} )}} \rbrack^{T}}$

According to the BPSK binary rank criterion, for C to achieve at least rlevels of diversity, the binary rank for each code word must be largerthan or equal to r. A code word matrix c has a binary rank equal to r,if and only if all matrices resulting from applying any number of simplerow operations to c have at least r non zero rows (i.e., the number ofzero rows is less than L_(t)−r+1). Noting that {Q(x, G)} is the set ofmatrices resulting from applying all possible combinations of simple rowoperations to the code word matrix corresponding to the input stream x,the following condition for a space-time code achieving r levels ofdiversity results:∀G∈G,x∈F^(K)└q ₁( x,G), q ₂( x,G), . . . , q _(L) _(t) _(−r+1)( x,G)┘≠0 _(1×n(L)_(t) _(−r+1))Now, the following relationships are used to obtain the generalized thestacking construction:

$\begin{matrix}{{q_{i}( {\underset{\_}{x},G} )} = {{\underset{\_}{x}\lbrack {{q_{i,1}I_{k}},{q_{i,2}I_{k}},\ldots\mspace{14mu},{q_{i,L_{t}}I_{k}}} \rbrack}\begin{bmatrix}M_{1} \\M_{2} \\\vdots \\M_{L_{t}}\end{bmatrix}}} \\{{= {\underset{\_}{x}\;{R_{i}(G)}}},}\end{matrix}$In the generalized stacking construction, C is assumed to be a linearL_(t)×n space-time code as defined in the previously described stackingconstruction with the exception that n≦k is allowed. Then, for BPSKtransmission over the quasi-static fading channel, C achieves at least dlevels of transmit diversity if d is the largest integer such that ∀G∈G,R(G)=└R₁(G), R₂(G), . . . , R_(L) _(t) _(−d+1)(G),┘ has a full rank kover the binary field F. It is clear that for d=L_(t), this conditionreduces to the previously described stacking construction. The abovegeneralized stacking construction allows for a significant increase inthroughput. For example, in the case of 4 transmit antennas 207, therate can be increased from 1 bits/sec/Hz to 2 bits/sec/Hz at the expenseof a small reduction in diversity advantage from 4 to 3.

The same idea behind the generalized stacking construction can be usedto design space-time codes for MIMO block fading channels. First, thefollowing notations are introduced:∀1≦m≦M,G _(m) =[g _(i,j)(m)]∈G,x∈F ^(K), let

$\begin{matrix}{{Q( {\underset{\_}{x},G_{m},m} )} = {G_{m}\begin{bmatrix}{\underset{\_}{x}\; M_{1m}} \\{\underset{\_}{x}\; M_{2m}} \\\vdots \\{\underset{\_}{x}\; M_{L_{t}m}}\end{bmatrix}}} \\{= \lbrack {{q_{1}^{T}( {\underset{\_}{x},G_{m},m} )},{q_{2}^{T}( {\underset{\_}{x},G_{m},m} )},\ldots\mspace{14mu},{q_{L_{t}}^{T}( {\underset{\_}{x},G_{m},m} )}} \rbrack^{T}}\end{matrix}$Using the block fading channel binary rank criterion and the generalizedstacking construction argument, it is observed that the space-time codeC achieves at least r levels of diversity if it satisfies the followingcondition:∀G∈G, . . . , G _(m) ∈G,x∈F ^(K), 0≦m ₁≦min(L _(t) ,ML _(t) −r+1), . . ., 0≦m _(M)≦min(L _(t) ,ML _(t) −r+1),and

${{\sum\limits_{m = 1}^{M}m_{i}} = {{ML}_{t} - r + 1}},$[q ₀( x,G ₁,1), . . . , q _(m) ₁ ( x,G ₁,1), q ₀( x,G ₂,2), . . . , q_(m) ₂ ( x,G ₂,2), . . . , q _(m) _(M) ( x,G _(M) ,M)]≠0_(1×n(L) _(t)_(−r+1))

where q₀(x,G_(m),m)=[ ] is an empty vector.

Again the following fact is utilized:

$\begin{matrix}{{q_{i}( {\underset{\_}{x},G_{m},m} )} = {{\underset{\_}{x}\lbrack {{{g_{i,1}(m)}I_{k}},{{g_{i,2}(m)}I_{k}},\ldots\mspace{14mu},{{g_{i,L_{t}}(m)}I_{k}}} \rbrack}\begin{bmatrix}M_{1m} \\M_{2m} \\\vdots \\M_{L_{t}m}\end{bmatrix}}} \\{{= {\underset{\_}{x}\;{R_{i}( {G_{m},m} )}}},}\end{matrix}$for i≧1 and R₀(G_(m),m) is an empty matrix, to obtain the result below,which applies the generalized stacking construction to MIMO block fadingchannels.

In the case of a block fading generalized stacking construction, C isassumed to be a linear L_(t)×n space-time code. For BPSK transmissionover a block fading channel 103 with M blocks, C achieves at least rlevels of transmit diversity if r is the largest integer such that∀G∈G, . . . , G _(m) ∈G,0≦m ₁≦min(L _(t) ,ML _(t) −r+1), . . . , 0≦m_(M)≦min(L _(t) ,ML _(t) −r+1), and

${{\sum\limits_{m = 1}^{M}m_{i}} = {{ML}_{t} - r + 1}},$

-   -   R_(m) ₁ _(, . . . , m) _(M) (G₁, . . . , G_(m))=└R₀(G₁), . . . ,        R_(m) ₁ (G₁), R₀(G₂), . . . , R_(m) ₂ (G₂), . . . , R_(m) _(M)        (G_(M))┘ has a full rank k over the binary field F.        It is straightforward to note that the multi-stacking        construction can be obtained by setting the diversity advantage        d=ML_(t). Further, it is interesting to compare the above        condition to the threaded stacking construction for layered        space-time systems.

Space-time layering is suitable for systems with relatively large numberof receive antennas. The large number of receive antennas allows for thedesign of efficient signal processing algorithms to separate the signalsfrom different layers efficiently, and hence, code design for layeredsystems can assume no spatial interference for between layers. Based onthis assumption, it is observed that the layered space-time code needsto satisfy the above condition when every ∀G_(m) is a permutation of theidentity matrix. This relaxed constraint constitute the base for theframework for layered space-time coding. For example, the presentinvention according to one embodiment, considers use of a relativelysmaller number of receive antennas 303 than the number of transmitantennas 207, thereby limiting the ability to separate the differenttransmit antenna signals. Hence, the burden now lays on the code designto account for the fact that the input to the maximum likelihood decoder305 is the sum of the different transmit antenna signals multiplied withthe corresponding fading coefficients.

The block fading generalized stacking construction permits constructionof space-time codes that optimally exploit the diversity available inMIMO block fading channels with arbitrary number of transmit antennas207 and fading blocks per code word. The special case of designingspace-time convolutional codes for this scenario is now described. Suchcodes advantageously enables the use of computationally efficientmaximum likelihood decoders (e.g., 305).

The natural space-time codes associated with binary rate 1/ML_(t)convolutional codes with periodic bit interleaving are attractivecandidates for the application under consideration in the presentinvention, as they can be easily formatted to satisfy the multi-stackingcondition, and hence achieve full diversity. Each output arm from theencoder 203 is assigned to a distinct pair of transmit antenna 207 andfading block.

The next step is to consider the more challenging task of finding ratek/ML_(t) codes that achieve the best trade-off between throughput anddiversity advantage for arbitrary choice of the transmission throughput.It is assumed that C is a binary convolutional code of rate k/ML_(t).The encoder 203 processes k binary input sequences x₁(t), x₂(t), . . . ,x_(k)(t) and produces ML_(t) coded output sequences y₁(t), y₂(t), . . ., y_(ML) _(t) (t) which are multiplexed together to form the output codeword. A sequence {x(t)} is often represented by the formal seriesX(D)=x(0) +x(1)D+x(2)D²+ . . . . {x(t)}

X(D) is referred to as a D-transform pair. The action of thebinary-convolutional encoder 203 is linear and is characterized by theso-called impulse responses g_(i,1)(t)

G_(i,j)(D) associating output y_(i)(t) with input x_(i)(t). Thus, theencoder 203 action is summarized by the matrix equation:Y(D)=X(D)G(D),where Y(D)=└Y ₁(D)Y ₂(D) . . . Y _(ML) _(t) (D)┘, X(D)=[X ₁(D)X ₂(D) . .. X _(k)(D)], and

${G(D)} = \begin{bmatrix}{G_{1,1}(D)} & {G_{1,2}(D)} & \cdots & {G_{1,{ML}_{t}}(D)} \\{G_{2,1}(D)} & {G_{2,2}(D)} & \cdots & {G_{1,{ML}_{t}}(D)} \\\vdots & \vdots & ⋰ & \vdots \\{G_{k,1}(D)} & {G_{k,2}(D)} & \cdots & {G_{k,{ML}_{t}}(D)}\end{bmatrix}$

The natural space-time formatting of C is considered, whereby the outputsequence corresponding to Y_((m−1)L) _(t) _(+l)(D) is assigned to thel^(th) transmit antenna 207 in the m^(th) fading block, such that thediversity that can be achieved by this scheme can be characterized. Thealgebraic analysis technique considers the rank of matrices formed byconcatenating linear combinations of the column vectors:

$\begin{matrix}{{F_{l}(D)} = \begin{bmatrix}{G_{1,l}(D)} \\{G_{2,l}(D)} \\\vdots \\{G_{k,l}(D)}\end{bmatrix}} & (15)\end{matrix}$Using the same approach as in the block fading generalized stackingconstruction, the following is defined ∀G_(m)∈G, 1≦i≦L_(t), 1≦m≦M

$\begin{matrix}{R_{i}^{({G_{m},m})}{\quad{(D) = {\lbrack {{{g_{i,1}(m)}I_{k}},{{g_{i,2}(m)}I_{k}},\ldots\mspace{14mu},{{g_{i,L_{t}}(m)}t_{k}}} \rbrack\begin{bmatrix}{F_{{{({m - 1})}L_{t}} + 1}(D)} \\{F_{{{({m - 1})}L_{t}} + 2}(D)} \\\vdots \\{F_{{mL}_{i}}(D)}\end{bmatrix}}}}} & (16)\end{matrix}$

An algebraic framework for designing BPSK space-time convolutional codesis now established. In a communication system 100 with L_(t) transmitantennas 207 operating over a block fading channel 103 with M blocks, Cdenotes the space-time code that includes the binary convolutional codeC, whose k×ML_(t) transfer function matrix is G(D)=└F₁(D) . . . F_(ML)_(t) (D)┘ and the spatial parser σ in which the output Y_((m−1)L) _(t)_(+l)(D)=X(D)F_((m−1)L) _(t) _(+l)(D) is assigned to antenna l in fadingblock m. Then, for BPSK transmission, C achieves d levels of transmitdiversity if d is the largest integer such that∀G ₁ ∈G, . . . , G _(M) ∈G, 0≦m ₁≦min(L _(t) ,ML _(t) −d+1), . . . , 0≦m_(M)≦min(L _(t) ,ML _(t) −r+1),

${{{and}\mspace{14mu}{\sum\limits_{m = 1}^{M}m_{i}}} = {{ML}_{t} - d + 1}},$R _(m) ₁ _(, . . . , m) _(M) ^((G) ¹ ^(, . . . , G) ^(M)) (D)=└R ₀ ^((G)¹ ^(,1))(D), . . . , R _(m) ₁ ^((G) ¹ ^(,1))(D), R ₀ ^((G) ² ^(,2))(D),. . . , R _(m) ₂ ^((G) ² ^(,2))(D), . . . , R _(mM) ^((G) ^(M)^(,M))(D)┘

has a rank k over the space of all formal series.

The above result is utilized to construct convolutional space-time codesthat realize the optimum tradeoff between transmission rate anddiversity order for BPSK modulation with arbitrary coding rate, numberof transmit antennas 207, and number of fading blocks. It can be easilyseen that this framework encompasses as special case rate 1/n′convolutional codes with bit or symbol interleaving across the transmitantennas and fading blocks.

The binary constructions previously proposed for BPSK modulation canalso be used to construct codes for higher order constellations. ForQPSK modulation, linear Z₄ codes can be constructed by “lifting” linearbinary codes to the Z₄ alphabet (i.e., the binary projection of the Z₄code generator matrix is the same as the binary generator matrix).Another way for designing codes for QPSK modulation is to combine twobinary codes A, B with the same diversity advantage in a dyadic formatC=A+2B. The resulting Z₄ code achieves the same diversity advantage asthe binary code, however, for QPSK transmission.

The present invention considers the design of space-time codes for MIMOblock fading channels. The developed baseband design criteria determinesthe diversity and coding advantage achieved by space-time codes in blockfading channels. For BPSK and QPSK modulated codes, the developed binaryrank criteria allow for designing space-time codes that exhibit theoptimum diversity-vs.- throughput tradeoff. These binary criteria arethen utilized to develop general algebraic framework for space-time codedesign in MIMO block fading channels. The above construct hasapplicability in a number of communication systems; for example, thespace-time codes can be deployed in a wireless communication, as seen inFIG. 5.

FIG. 5 shows a diagram of a wireless communication system that utilizesspace-time codes, according to an embodiment of the present invention.In a wireless communication system 500, multiple terminals 501 and 503communicate over a wireless network 505. Terminal 501 is equipped with aspace-time encoder 203 (as shown in FIG. 2) that generates space-timecodes with a block fading generalized stacking construction. Terminal501 also includes multiple transmit antennas 207 (as shown in FIG. 2).In this example, each of the terminals 501 and 503 are configured toencode and decode the space-time codes; accordingly, both of theterminals 501 and 503 possess the transmitter 200 and receiver 300.However, it is recognized that each of the terminals 501 and 503 mayalternatively be configured as a transmitting unit or a receiving unit,depending on the application. For example, in a broadcast application,terminal 501 may be used as a head-end to transmit signals to multiplereceiving terminals (in which only receiving terminal 503 is shown).Consequently, terminal 503 would only be equipped with a receiver 300.The space-time code construction of the present invention advantageouslypermits use of a smaller number of receive antennas than that of thetransmitting terminal 501, thereby resulting in hardware cost reduction.In an exemplary embodiment, a terminal that is designated as a receivingunit may possess a smaller quantity of antennas that of the transmittingunit.

FIG. 6 shows a diagram of a computer system that can perform theprocesses of encoding and decoding of space-time codes, in accordancewith an embodiment of the present invention. Computer system 601includes a bus 603 or other communication mechanism for communicatinginformation, and a processor 605 coupled with bus 603 for processing theinformation. Computer system 601 also includes a main memory 607, suchas a random access memory (RAM) or other dynamic storage device, coupledto bus 603 for storing information and instructions to be executed byprocessor 605. In addition, main memory 607 may be used for storingtemporary variables or other intermediate information during executionof instructions to be executed by processor 605. Computer system 601further includes a read only memory (ROM) 609 or other static storagedevice coupled to bus 603 for storing static information andinstructions for processor 605. A storage device 611, such as a magneticdisk or optical disk, is provided and coupled to bus 603 for storinginformation and instructions.

Computer system 601 may be coupled via bus 603 to a display 613, such asa cathode ray tube (CRT), for displaying information to a computer user.An input device 615, including alphanumeric and other keys, is coupledto bus 603 for communicating information and command selections toprocessor 605. Another type of user input device is cursor control 617,such as a mouse, a trackball, or cursor direction keys for communicatingdirection information and command selections to processor 605 and forcontrolling cursor movement on display 613.

According to one embodiment, channel code generation within system 100is provided by computer system 601 in response to processor 605executing one or more sequences of one or more instructions contained inmain memory 607. Such instructions may be read into main memory 607 fromanother computer-readable medium, such as storage device 611. Executionof the sequences of instructions contained in main memory 607 causesprocessor 605 to perform the process steps described herein. One or moreprocessors in a multi-processing arrangement may also be employed toexecute the sequences of instructions contained in main memory 607. Inalternative embodiments, hard-wired circuitry may be used in place of orin combination with software instructions. Thus, embodiments are notlimited to any specific combination of hardware circuitry and software.

Further, the instructions to support the generation of space-time codesof system 100 may reside on a computer-readable medium. The term“computer-readable medium” as used herein refers to any medium thatparticipates in providing instructions to processor 605 for execution.Such a medium may take many forms, including but not limited to,non-volatile media, volatile media, and transmission media. Non-volatilemedia includes, for example, optical or magnetic disks, such as storagedevice 611. Volatile media includes dynamic memory, such as main memory607. Transmission media includes coaxial cables, copper wire and fiberoptics, including the wires that comprise bus 603. Transmission mediacan also take the form of acoustic or light waves, such as thosegenerated during radio wave and infrared data communication.

Common forms of computer-readable media include, for example, a floppydisk, a flexible disk, hard disk, magnetic tape, or any other magneticmedium, a CD-ROM, any other optical medium, punch cards, paper tape, anyother physical medium with patterns of holes, a RAM, a PROM, and EPROM,a FLASH-EPROM, any other memory chip or cartridge, a carrier wave asdescribed hereinafter, or any other medium from which a computer canread.

Various forms of computer readable media may be involved in carrying oneor more sequences of one or more instructions to processor 605 forexecution. For example, the instructions may initially be carried on amagnetic disk of a remote computer. The remote computer can load theinstructions relating to encoding and decoding of space-time codes usedin system 100 remotely into its dynamic memory and send the instructionsover a telephone line using a modem. A modem local to computer system601 can receive the data on the telephone line and use an infraredtransmitter to convert the data to an infrared signal. An infrareddetector coupled to bus 603 can receive the data carried in the infraredsignal and place the data on bus 603. Bus 603 carries the data to mainmemory 607, from which processor 605 retrieves and executes theinstructions. The instructions received by main memory 607 mayoptionally be stored on storage device 611 either before or afterexecution by processor 605.

Computer system 601 also includes a communication interface 619 coupledto bus 603. Communication interface 619 provides a two-way datacommunication coupling to a network link 621 that is connected to alocal network 623. For example, communication interface 619 may be anetwork interface card to attach to any packet switched local areanetwork (LAN). As another example, communication interface 619 may be anasymmetrical digital subscriber line (ADSL) card, an integrated servicesdigital network (ISDN) card or a modem to provide a data communicationconnection to a corresponding type of telephone line. Wireless links mayalso be implemented. In any such implementation, communication interface619 sends and receives electrical, electromagnetic or optical signalsthat carry digital data streams representing various types ofinformation.

Network link 621 typically provides data communication through one ormore networks to other data devices. For example, network link 621 mayprovide a connection through local network 623 to a host computer 625 orto data equipment operated by a service provider, which provides datacommunication services through a communication network 627 (e.g., theInternet). LAN 623 and network 627 both use electrical, electromagneticor optical signals that carry digital data streams. The signals throughthe various networks and the signals on network link 621 and throughcommunication interface 619, which carry the digital data to and fromcomputer system 601, are exemplary forms of carrier waves transportingthe information. Computer system 601 can transmit notifications andreceive data, including program code, through the network(s), networklink 621 and communication interface 619.

The techniques described herein provide several advantages over priorapproaches to providing space-time codes for MIMO block fading channels.Such code construction enhances spatial and temporal diversity withoutsacrificing transmission rate.

Obviously, numerous modifications and variations of the presentinvention are possible in light of the above teachings. It is thereforeto be understood that within the scope of the appended claims, theinvention may be practiced otherwise than as specifically describedherein.

REFERENCES

[1] V. Tarokh, N. Seshadri, and A. R. Calderbank. Space-time codes forhigh data rate wireless communication: Performance criterion and codeconstruction. IEEE Trans. Info. Theory, IT-44:774–765, March 1998.

[2] J.-C. Guey, M. R. Bell M. P. Fitz, and W.-Y. Kuo. Signal design fortransmitter diversity wireless communication systems over Rayleighfading channels. IEEE Vehicular Technology Conference, pages 136–140,Atlanta, 1996.

[3] E. Biglieri, G. Caire, and G. Taricco. Limiting performance forblock-fading channels with multiple antennas, submitted to IEEE Trans.Info. Theory, September 1999.

1. A method for generating space-time codes, the method comprising:computing a baseband design criteria for determining a diversityadvantage and a coding advantage achieved over block fading channels;translating the baseband design criteria into a binary rank criteria formodulated codes corresponding to the space-time codes; and outputtingthe space-time codes according an algebraic framework derived from thebinary rank criteria, wherein the binary rank criteria specifies thatfor every non-zero code word c∈C,${{\sum\limits_{m = 1}^{M}{{rank}( {c\lbrack m\rbrack} )}} \geq d},$ the rank is over a binary field F and d is the largest possibleinteger, the space-time code C achieving a diversity level of at leastdL_(n) L_(r) being the number of the plurality of receive antennas, mrepresenting the number of blocks per code word.
 2. A method accordingto claim 1, wherein the transmit diversity advantage is as follows:$d_{m} = {\sum\limits_{m = 1}^{M}{{rank}( {{f( {c\lbrack m\rbrack} )} - {f( {e\lbrack m\rbrack} )}} )}}$over all pairs of distinct code words c, e∈C, m representing a fadingblock.
 3. A method according to claim 1, wherein the coding advantage isas follows:$\mu = ( {\prod\limits_{m = 1}^{M}{{\lambda_{1}\lbrack m\rbrack}{\lambda_{2}\lbrack m\rbrack}\mspace{14mu}\cdots\mspace{14mu}{\lambda_{d_{m}}\lbrack m\rbrack}}} )^{1/d}$over all pairs of distinct code words c, e∈C, m representing a fadingblock, λ₁[m]λ₂[m] . . . λ_(d) _(m) [m] being nonzero elgenvalues of(f(c[m])−f(e[m]))(f(c[m])−(e[m]))^(H).
 4. A method according to claim 1,wherein the algebraic framework specifies C as a linear L_(t)×nspace-time code, the space-time code being of dimension k including codeword matrices, $c = \begin{bmatrix}{\underset{\_}{x}M_{1}} \\{\underset{\_}{x}M_{2}} \\\vdots \\{\underset{\_}{x}M_{L_{t}}}\end{bmatrix}$ wherein x denotes an arbitrary k-tuple of informationbits and n≧L_(t), M₁, M₂, . . . , M_(L) _(t) are binary matrices ofdimension k×n, n≦k, and L_(t) represents the number of transmit antennasin the communication system.
 5. A method according to claim 4, whereinM₁, M₂, . . . , M_(L) _(t) has the following property,∀a₁, a₂, . . . , a_(L) _(t) ∈F:M=a₁M₁⊕a₂M₂⊕ . . . ⊕a_(L) _(t) M_(L) _(t) is of full rank k unlessa₁=a₂= . . . =a_(L) _(t) =0.
 6. A method for generating space-timecodes, the method comprising: computing a baseband design criteria fordetermining a diversity advantage and a coding advantage achieved overblock fading channels; translating the baseband design criteria into abinary rank criteria for modulated codes corresponding to the space-timecodes; and outputting the space-time codes according an algebraicframework derived from the binary rank criteria, wherein the binary rankcriteria specifies for every non-zero code word c∈C, at least one of${{\sum\limits_{m = 1}^{M}\;{{rank}\{ {\Xi( {c\lbrack m\rbrack} )} \}}} \geq d},{{{and}\mspace{14mu}{\sum\limits_{m = 1}^{M}\;{{rank}\{ {\Psi( {c\lbrack m\rbrack} )} \}}}} \geq d}$ holds true, wherein the rank is over a binary field and d is thelargest possible integer, Ξ being a row-based indicant projection, Ψbeing a column-based indicant projection, the space-time code Cachieving a diversity level of at least dL_(n) L_(r) being the number ofthe plurality of receive antennas, m representing the number of blocksper code word.
 7. An apparatus for generating space-time codes, theapparatus comprising: means for computing a baseband design criteria fordetermining a diversity advantage and a coding advantage achieved overblock fading channels; means for translating the baseband designcriteria into a binary rank criteria for modulated codes correspondingto the space-time codes; and means for outputting the space-time codesaccording an algebraic framework derived from the binary rank criteria,wherein the binary rank criteria specifies that for every non-zero codeword c∈C,${{\sum\limits_{m = 1}^{M}{{rank}( {c\lbrack m\rbrack} )}} \geq d},$ the rank is over a binary field F and d is the largest possibleinteger, the space-time code C achieving a diversity level of at leastdL_(n) L_(r) being the number of the plurality of receive antennas, mrepresenting the number of blocks per code word.
 8. An apparatusaccording to claim 7, wherein the transmit diversity advantage is asfollows:$d_{m} = {\sum\limits_{m = 1}^{M}{{rank}( {{f( {c\lbrack m\rbrack} )} - {f( {e\lbrack m\rbrack} )}} )}}$ over all pairs of distinct code words c, e∈C, m representing a fadingblock.
 9. An apparatus according to claim 7, wherein the codingadvantage is as follows:$\mu = ( {\prod\limits_{m = 1}^{M}{{\lambda_{1}\lbrack m\rbrack}{\lambda_{2}\lbrack m\rbrack}\mspace{14mu}\cdots\mspace{14mu}{\lambda_{d_{m}}\lbrack m\rbrack}}} )^{1/d}$over all pairs of distinct code words c, e∈C, m representing a fadingblock, λ₁[m]λ₂[m] . . . λ_(d) _(m) [m] being nonzero eigenvalues of(f(c[m])−f(e[m]))(f(c[m])−f(e[m]))^(H).
 10. An apparatus according toclaim 7, wherein the algebraic framework specifies C as a linear L_(t)×nspace-time code, the space-time code being of dimension k including codeword matrices, $c = \begin{bmatrix}{\underset{\_}{x}M_{1}} \\{\underset{\_}{x}M_{2}} \\\vdots \\{\underset{\_}{x}M_{L_{t}}}\end{bmatrix}$ wherein x denotes an arbitrary k-tuple of informationbits and n≧L_(t), M₁, M₂, . . . , M_(L) _(t) are binary matrices ofdimension k×n, n≦k, and L_(t) represents the number of transmit antennasin the communication system.
 11. An apparatus according to claim 10,wherein M₁, M₂, . . . , M_(L) _(t) has the following property,∀a₁, a₂, . . . , a_(L) _(t) ∈F:M=a₁M₁⊕a₂M₂⊕ . . . ⊕a_(L) _(t) M_(L) _(t) is of full rank k unlessa₁=a₂= . . . =a_(L) _(t) =0. 3
 12. An apparatus according to claim 7,wherein the binary rank criteria specifies for every non-zero code wordc∈C, at least one of${{\sum\limits_{m = 1}^{M}\;{{rank}\{ {\Xi( {c\lbrack m\rbrack} )} \}}} \geq d},{{{and}\mspace{14mu}{\sum\limits_{m = 1}^{M}\;{{rank}\{ {\Psi( {c\lbrack m\rbrack} )} \}}}} \geq d}$holds true, wherein the rank is over a binary field and d is the largestpossible integer, Ξ being a row-based indicant projection, Ψ being acolumn-based indicant projection, the space-time code C achieving adiversity level of at least dL_(n) L_(r) being the number of theplurality of receive antennas, m representing the number of blocks percode word.
 13. A method for communicating over a communication system,the method comprising: generating a code word in response to an inputmessage, the code word being based upon a stacking constructiongeneralized for a multi-input multi-output (MIMO) block fading channel,wherein the generalized stacking construction specifies C as a linearL_(t)×n space-time code, the space-time code being of dimension kincluding code word matrices, $c = \begin{bmatrix}{\underset{\_}{x}M_{1}} \\{\underset{\_}{x}M_{2}} \\\vdots \\{\underset{\_}{x}M_{L_{t}}}\end{bmatrix}$  wherein x denotes an arbitrary k-tuple of informationbits and n≧L_(t), M₁, M₂, . . . , M_(L) _(t) are binary matrices ofdimension k×n, n≦k, and L_(t) represents the number of transmit antennasin the communication system; and transmitting the code word over aplurality of transmit antennas to a receiver configured to decode thecode word using maximum likelihood decoding, wherein the receiverincludes a plurality of receive antennas.
 14. A method according toclaim 13, further comprising: modulating the code word based on eitherBPSK (binary phase-shift keying) modulation or QPSK (quadraturephase-shift keying) modulation.
 15. A method according to claim 13,wherein the number of receive antennas is less than the number oftransmit antennas.